Relative efficiency of the MLE and Method of Moments Estimator

Hello I am trying determine which estimator is a more efficient estimator for the parameter $\theta$, the methods of moments estimator or maximum likelihood estimator for the distribution of $$f(x;\theta) = \frac 1 \theta \left(\frac 1 x\right)^{\frac 1 \theta + 1}$$ for $x > 1$, $0 < \theta < 1$.

Using the first moment I found that $\bar{x} = \frac{1}{1 - \theta}$ which implies the method of moments estimator is $\hat\theta_1 = 1 - \frac 1 {\bar{x}}$. I also found the maximum likelihood estimator to be $\hat\theta_2 =\bar{y}$ for $y = \ln x_i$ (the maximum likelihood estimator I found was just the mean of the natural logarithms of all the $x_i$ in this distribution). I also found that the estimator $\hat\theta_1$ is biased and $\hat\theta_2$ is unbiased and happens to be a minimum variance unbiased estimator of $\theta$. By definition a parameter estimator is supposed to be relatively more efficient if it has a smaller variance than the other estimator and by the Cramer-Rao lower bound theorem I know that the maximum likelihood estimator $\hat\theta_2$ in this distribution achieves the smallest possible variance in all of the unbiased estimators.

The difficulty I am having in determining which of these two estimators is more efficient (has a smaller variance) is computing the variance for the method of moments estimator $\hat\theta_1 = 1 - \frac{1}{\bar{x}}$, and because this estimator is biased I believe it is possible for it to have a smaller variance than the maximum likelihood estimator $\hat\theta_2$. Is there any way to determine which of these two estimators is a relatively more efficient estimator of the parameter $\theta$ without having to compute the variance of the biased estimator $\hat\theta_1$? And if not, how would I go about computing the variance for $\hat\theta_1 = 1 - \frac{1}{\bar{x}}$?

• Two thoughts: (1) If you showed that the UMVUE $\hat{\theta}_1$ attains the Cramer-Rao lower bound, then the CRLB is the variance. I realize this might be unlikely that you showed this - but just throwing it in there. Also, for exponential families, there might be conditions such that if satisfied the UMVUE estimator will reach the bound. (2) If you are looking at asymptotic efficiency, you may consider using the delta-method to identify the variance of the asymptotic distribution of $\hat{\theta}_1$ – Just_to_Answer May 7 '17 at 2:54
• Instead of the variance of the biased estimator, are you sure you don't want the mean squared error? That is equal to the variance plus the square of the bias. $\qquad$ – Michael Hardy May 7 '17 at 3:23